Answer:
a) Fill in the spaces
The probability of the event "have a Bachelor's Degree" is affected by the occurrence of the event "never married", and the probability of the event "never married" is affected by the occurrence of the event "have a Bachelor's Degree", so the events are not independent.
b) Probability of a woman aged 25 or older having a bachelor's degree and having never married = P(B n NM) = 0.0369
This probability is the probability of the intersect of the two events, 'have bachelor's degree' and 'have never married' for women aged 25 or older.
Step-by-step explanation:
Complete Question
According to a government statistics department, 20.6% of women in a country aged 25 years or older have a Bachelor's Degree; 16.6% of women in the country aged 25 years or older have never married; among women in the country aged 25 years or older who have never married, 22.2% have a Bachelor's Degree; and among women in the country aged 25 years or older who have a Bachelor's Degree, 17.9% have never married. Complete parts a) and (b) below.
(a) Are the events "have a Bachelor's Degree" and "never married"? independent? Explain.
(b) Suppose a woman in the country aged 25 years or older is randomly selected. What is the probability she has a Bachelor's Degree and has never married? Interpret this probability.
Solution
The probability of the event that a woman aged 25 or older has a bachelor's degree = P(B) = 20.6% = 0.206
The probability of the event that a woman aged 25 or older has never married = P(NM) = 16.6% = 0.166
Among women in the country aged 25 years or older who have never married, 22.2% have a Bachelor's Degree.
This means that the probability of having a bachelor's degree given that a woman aged 25 or older have never married is 22.2%.
P(B|NM) = 22.2% = 0.222
And among women in the country aged 25 years or older who have a Bachelor's Degree, 17.9% have never married
This means that the probability of having never married given that a woman aged 25 or older has bachelor's degree is 22.2
P(NM|B) = 17.9% = 0.179
a) To investigate if the two events 'have a bachelor's degree' and 'have never married' are independent for women aged 25 or older.
Two events are said to be independent if the probability of one of them occurring does not depend on the probability of the other occurring. Two events A and B can be proven mathematically to be independent if
P(A|B) = P(A) or P(B|A) = P(B)
For the two events in question,
P(B) = 0.206
P(NM) = 0.166
P(B|NM) = 0.222
P(NM|B) = 0.179
It is evident that
P(B) = 0.206 ≠ 0.222 = P(B|NM)
P(NM) = 0.166 ≠ 0.179 = P(NM|B)
Since the probabilities of the two events do not satisfy the conditions for them to be independent, the two events are not independent.
b) Probability of a woman aged 25 or older having a bachelor's degree and having never married = P(B n NM)
The conditional probability, P(A|B), is given mathematically as
P(A|B) = P(A n B) ÷ P(B)
P(A n B) = P(A|B) × P(B)
or
= P(B|A) × P(A)
Hence,
P(B n NM) = P(NM n B) = P(B|NM) × P(NM) = P(NM|B) × P(B)
P(B|NM) × P(NM) = 0.222 × 0.166 = 0.036852 = 0.0369
P(NM|B) × P(B) = 0.179 × 0.206 = 0.036874 = 0.0369
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